Mean First-Passage Time of an Asymmetric Kinetic Model Driven by Two Different Kinds of Colored Noises

The mean first-passage time （MFPT） of an asymmetric bistable system between multiplicative non-Gaussian noise and additive Gaussian white noise with nonzero cross-correlation time is investigated. Firstly, the non-Markov process is reduced to the Markov process through a path-integral approach; Secondly, the approximate Fokker Planck equation is obtained by applying the unified colored noise approximation and the.Novikov Theorem. The steady-state probability distribution （SPD） is also obtained. The basal functional analysis and simplification are employed to obtain the approximate expressions of MFPT T^±. The effects of the asymmetry parameter β, the non-Gaussian parameter （measures deviation from Gaussian character） r, the noise correlation times τ and τ2, the coupling coefficient A, the intensities D and a of noise on the MFPT are discussed. It is found that the asymmetry parameter β, the non-Gaussian parameter r and the coupling coefficient A can induce phase transition. Moreover, the main findings are that the effect of self-existent parameters （D, α, and τ） of noise and cross-correlation parameters （A, 7-2） between noises on MFPT T^± is different.     

Moments and Mean First—Passage Time of Parabolic—Bistable Potential System Driven by Colored Noise

A parabolic-bistable potential system driven by colored noise is studied.The exact analytical expressions of the stationary probability distribution (SPD) and the moments of the system are derived.Furthermore,the mean first-passage time is calculated by the use of two approximate methods,respectively.It is found that (i) the double peaks of SPD are rubbed-down into a flat single peak with the increasing of noise intensity;(ii) a minimum occurs on the curve of the second-order moment of the system vs.noise intensity at the point DΓ=0.025;(iii) the results obtained by our approximate approach are in good agreement with the numerical calculations for either small or large correlation time τ,while the conventional steepest descent approximation leads to poor results.     

Stochastic properties coupling between of tumor growth with non-Gaussian and Gaussian noise terms

Dynamical behavior of a tumor-growth model with coupling between non-Gaussian and Gaussian noise terms is investigated. The departure from the Gaussian noise can not only reduce the probability of tumor cells in the active state, induce the minimum of the average tumor-cell population to move toward a smaller non-Gaussian noise, but also decrease the mean first-passage time. The increase of white-noise intensity can increase the tumor-cell population and shorten the mean first-passage time, while the coupling strength between noise terms has opposite effects, and the noise correlation time has a very small effect.     

Mean First—Passage Time of a Bistable Kinetic Model Driven by Multiplicative Coloured Noise and Additive White Noise

We study the transient properties of a bistable kinetic system driven by correlated noises for the cases of multiplicative coloured noise and additive white noise. The mean first-passage time (MFPT) of the system is calculated. From numerical computations we find that: (i) the MFPT of the bistable system are affected by the correlation time of multiplicative coloured noise T and the cross-correlation strength between noises λ and, T and λ play the same roles in the MFPT. (ii) The MFPT corresponding to weakly correlated noises and strongly correlated noises exhibits the very different behaviour and there is a one-peak structure in the MFPT for strongly correlated noises. The peak grows highly as T increases, which means that the noisy colour causes the suppression effect of the escape rate to become more pronounced.     

Resonant-Like Activation in a Bistable System with Noise and Time Delay

A bistable system with noise and time delay is investigated. Theoretical analysis and stochastic simulation show that： （i） In the ease of a system driven only by multiplicative Gaussian white noise, the mean first-passage time for a particle to reach the other stable state from one stable state exhibits a minimum with respect to delay time, i.e., a resonant-like activation （RA） phenomenon. （ii） In the action of additive and multiplicative noise, as the additive noise intensity increases, no matter whether a correlation between the two types of noise exists or not, the RA gradually disappears. （iii） The correlation strength between the two types of noise does not influence the existence of the RA.     

Stochastic Resonance in a Bistable System Subject to Non-Gaussian and Gaussian Noises

In this paper, we consider the phenomenon of stochastic resonance （SR） in a quartic bistable system under the simultaneous action of a multiplicative non-Gaussian and an additive Gaussian noises and a weak periodic signal. The expression of the signal-to-noise ratio R is derived by applying the two-state theory in adiabatic limit. We discuss the effects of the parameter q indicating the departure of the non-Gaussian noise from the Gaussian noise, the correlation time r of the non-Gaussian noise, and coupling intensity A between two noise terms on the stochastic resonance. It is found that the signM-to-noise ratio of the system, as a function of the additive noise intensity, undergoes the transition from having one peak to having two peaks, and then to having one peak again when the parameter q or the noise correlation time τ is increased. The parameter q and τ play opposite roles in the SR of the system.     

Intensity Correlation Time of a Single-Mode Laser Driven by Two Coloured Noises with Coloured Cross-Correlation with Bias Signal Modulation

The intensity correlation time T is studied by employing a gain-noise model of a single-mode laser driven by coloured pump noise τ1 and coloured quantum noise τ2 with coloured cross-correlation τ3 with a bias signal modulation. By using the linear approximation method, we detect that a maximum （i.e. resonance） exists in the curves of the intensity correlation time T versus the noise intensities D and Q when the noise correlation coefficient A is positive; and a minimum （i.e. suppression） exists in the T - D and T - Q curves when A is negative. When A is zero, T increases monotonously with increase of D and decreases monotonically with increase of Q. Furthermore, the curve of T versus the pump noise self-correlation time τ1 is also studied. Our study shows that, no matter what the value of A is, there exist one maximum and one minimum in the T - τ1 curve.     

Effect of Asymmetric Potential and Gaussian Colored Noise on Stochastic Resonance

The phenomenon of stochastic resonance （SR） in a bistable nonlinear system is studied when the system is driven by the asymmetric potential and additive Gaussian colored noise. Using the unified colored noise approximation method, the additive Gaussian colored noise can be simplified to additive Gaussian white noise. The signal-to-noise ratio （SNR） is calculated according to the generalized two-state theory （shown in [H.S. Wio and S. Bouzat, Brazilian J.Phys. 29 （1999） 136]）. We find that the SNR increases with the proximity of a to zero. In addition, the correlation time T between the additive Gaussian colored noise is also an ingredient to improve SR. The shorter the correlation time T between the Gaussian additive colored noise is, the higher of the peak value of SNR.     

Effect of Time Delay on Binary Signal Detection via a Bistable System

The effect of time delay on binary signal detection via a bistable system in the presence of white or colored Gaussian noise is investigated. By defining the bit error rate based on the solution of the approximated Fokker- Planck equation, the detector performance is investigated theoretically and is verified by Monte Carlo simulation. It is shown that, when the system parameter or noise intensity is optimally chosen, the increasing time delay generally improves the system performance. It is also shown that it is more difficult to accurately predict the system performance with a larger time delay and correlation time. This may inspire more thorough investigations in cooperative effects of a nonlinear system and time delay on signal processing.     

Mean first-passage time of an asymmetric bistable system driven by colour-correlated noise

In this paper, the effect of every parameter (including p, q, r, \la, \tau) on the mean first-passage time (MFPT) is investigated in an asymmetric bistable system driven by colour-correlated noise. The expression of MFPT has been obtained by applying the steepest-descent approximation. Numerical results show that (1) the intensity of multiplicative noise p and the intensity of additive noise q play different roles in the MFPT of the system, (2) suppression appears on the curve of the MFPT with small \la (e.g. \la<0.5) but there is a peak on the curve of the MFPT when \la is big (e.g. \la >0.5), and (3) with different values of r (e.g. r=0.1, 0.5, 1.5), the effort of \tau on the MFPT is diverse.     

Effect of Asymmetric Potential and Gaussian Colored Noise on Stochastic Resonance

The phenomenon of stochastic resonance (SR) in a bistable nonlinear system is studied when the system is driven by the asymmetric potential and additive Gaussian colored noise. Using the unified colored noise approximation method, the additive Gaussian colored noise can be simplified to additive Gaussian white noise. The signal-to-noise ratio (SNR) is calculated according to the generalized two-state theory (shown in [H.S. Wio and S. Bouzat, Brazilian J.Phys. 29 (1999) 136]). We find that the SNR increases with the proximity of a to zero. In addition, the correlation time τ between the additive Gaussian colored noise is also an ingredient to improve SR. The shorter the correlation time τ between the Gaussian additive colored noise is, the higher of the peak value of SNR.     

Enhancement of density divergence in an insect outbreak model driven by colored noise

The steady states and the transient properties of an insect outbreak model driven by Gaussian colored noise are studied in this paper.According to the Fokker-Planck equation in the unified colored-noise approximation,we analyse the stationary probability distribution and the mean first-passage time of this model.By numerical analysis,the effects of the self-correlation time of insect birth rate and predation rate respectively reveal a manifest population divergence on the insect density.The decrease of the mean first-passage time indicates an enhancement dynamic on the density divergency with colored noise of a large self-correlation time based on the insect outbreak model.     

The mean first passage time of a three-level atomic optical bistable system subjected to noise

The transient properties of a three-level atomic optical bistable system in the presence of multiplicative and additive noises are investigated. The explicit expressions of the mean first-passage time (MFPT) of the transition from the high intracavity intensity state to the low one are obtained by numerical computations. The impacts of the intensities of the multiplicative noise D M and the additive noise D A , the intensity of correlation between two noises λ , and the intensity of the incident light y on the MFPT are discussed, respectively. Our results show: (i) for the case of no correlation between two noises (λ = 0.0), the increase in D M and D A can lead to an increase in the probability of the transition to the low intracavity intensity state, while the increase in y can lead to a retardation of the transition; and (ii) for the case of correlation between two noises (λ = 0.0), the increase in λ can cause an increase in the probability of the transition, and the increase in D A can cause a retardation of the transition firstly and then an increase in the probability of the transition, i.e., the noise-enhanced stability is observed for the case of correlation between two noises.     

Effects of non-Gaussian noise on a calcium oscillation system

We investigate the effects of the non-Gaussian colored noise on a calcium oscillation system using stochastic simulation methods. It is found that the reciprocal coefficient of variance R has a maximum (R max ) with increasing noise intensity Q. The non-Gaussian noise parameter q has an important effect on the system. For some values of q (e.g., q = 0.9, q = 1.0), R has a maximum with increasing correlation time τ. Non-Gaussian noise induced spikes are more regular than Gaussian noise induced spikes when q is small and Q has large values. The R has a maximum with increasing q. Therefore, non-Gaussian noise could play more effective roles in the calcium oscillation system.     

Stochastic resonance in a bistable system with coloured correlation between additive and multiplicative noise

The phenomenon of stochastic resonance (SR) in a bistable nonlinear system with coupling between additive and multiplicative noises is investigated when the correlation between two noise terms is coloured. It is found that the signal-to-noise ratio (SNR) of the system is affected not only by the coupling strength λ between two noise terms, but also by the noise correlation time τ. The SNR is changed from a single peak, to two peaks with a dip, and then to a monotonically decreasing function with noise strength. The dependence of the SR on the initial conditions is entirely caused by the coupling strength λ between two noise terms.     

Colored Noise in First-order-like Phase Transition of a Laser System

The decoupling theory is employed to analyze the multiplicative colored noise in a single mode laser system. Steady state intensity distribution function is derived when colored noise is included in the laser system. The first-order-like phase transition driven by multiplicative colored noise is investigated and compared with the case of multiplicative white noise. It is shown that the noise correlation time can affect the parameter plane of the first-order-like phase transition. The steady state intensity distributions in a laser system is changed greatly with noise correlation time τ.     

Extinction Time of a Metapopulation Driven by Colored Correlated Noises

The simplified incidence function model which is driven by the colored correlated noises is employed to investigate the extinction time of a metapopulation perturbed by environments. The approximate Fokker-Planck Equation and the mean first passage time which denotes the extinction time （Tex） are obtained by virtue of the Novikov theorem and the Fox approach. After introducing a noise intensity ratio and a dimensionless parameter R = D /α （D and a are the multiplicative and additive colored noise intensities respectively）, and then performing numerical computations, the results indicate that： （i） The absolute value of correlation strength A and its correlation time τ3 play opposite roles on the Tex; （ii） For the case of 0 〈λ〈 1,α and its correlation time τ2 play opposite roles on the Tex in which R〉 1 is the best condition, and there is one-peak structure on the Tex - D plot; （iii） For the case of-1 〈 λ≤ 0, D and its correlation time τ1 play opposite roles on the Tex in which R 〈 1 is the best condition and there is one-peak structure on the Tex - τ2 plot.     

Ballistic diffusion induced by non-Gaussian noise

In this letter,we have analyzed the diffusive behavior of a Brownian particle subject to both internal Gaussian thermal and external non-Gaussian noise sources.We discuss two time correlation functions C(t) of the non-Gaussian stochastic process,and find that they depend on the parameter q,indicating the departure of the non-Gaussian noise from Gaussian behavior:for q ≤ 1,C(t) is fitted very well by the first-order exponentially decaying curve and approaches zero in the longtime limit,whereas for q 1,C(t) can be approximated by a second-order exponentially decaying function and converges to a non-zero constant.Due to the properties of C(t),the particle exhibits a normal diffusion for q ≤ 1,while for q 1 the non-Gaussian noise induces a ballistic diffusion,i.e.,the long-time mean square displacement of the free particle reads [x(t)-x(t)]2∝t2.     

Random walks in generalized delayed recursive trees

Recently a great deal of effort has been made to explicitly determine the mean first-passage time （MFPT） between two nodes averaged over all pairs of nodes on a fractal network. In this paper, we first propose a family of generalized delayed recursive trees characterized by two parameters, where the existing nodes have a time delay to produce new nodes. We then study the MFPT of random walks on this kind of recursive tree and investigate the effect of the time delay on the MFPT. By relating random walks to electrical networks, we obtain an exact formula for the MFPT and verify it by numerical calculations. Based on the obtained results, we further show that the MFPT of delayed recursive trees is much shorter, implying that the efficiency of random walks is much higher compared with the non-delayed counterpart. Our study provides a deeper understanding of random walks on delayed fractal networks.     

Entropic Stochastic Resonance Driven by Colored Noise

The phenomenon of entropic stochastic resonance （ESR） in a two-dimensional confined system driven by a transverse periodic force is investigated when the colored fluctuation is included in the system. Applying the method of unified colored noise approximation, the approximate Fokker-Planck equation can be derived in the absence of the periodic force. Through the escaping rate of the Brownian particle from one well to the other, the power spectral amplification can be obtained. It is found that increasing the values of the noise correlation time and the signal frequency can suppress the ESR of the system.